Finite Group Databasebeta

← #190213

Subgroups — #190213

Order 512.

1110 conjugacy classes of subgroups.

The subgroup lattice has 417 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34184C211yesmaximal, characteristic
2#20632C211yesmaximal, characteristic
3#17349C211yesmaximal, characteristic
4#8761C218yesmaximal, complemented
5#8026C212yesmaximal, complemented
6#7981C212yesmaximal, complemented
7C2×#520—21no—
8C2×C4×#41C2211yescharacteristic
9C2×#431C2212yes—
10C2×#410—21no—
11#2351—21no—
12#2351C412yescomplemented
13#2349—21no—
14#2349—22no—
15#2349C412yescomplemented
16#2349C2214yes—
17#2317—22no—
18#2317—21no—
19#2285C412yescomplemented
20#2285C2212yes—
21#2285C2211yescharacteristic
22C2×#382C412yescomplemented
23C2×#382C2211yescharacteristic
24#1799—22no—
25#1799C2214yes—
26#1740—21no—
27#1461—21no—
28#1460—21no—
29#1459C2214yes—
30#1458—22no—
31#1294C2218yes—
32#1289C2218yes—
33C22×#41D411yescharacteristic
34#520—48no—
35C4×#41—22no—
36C4×#41—22no—
37C4×#41—22no—
38C4×#41—22no—
39C2×C4×Q8—22no—
40C2×C4×Q8C2×C411yescharacteristic
41C2×C4×Q8D411yescharacteristic
42C2×C4×D4—22no—
43C2×C4×D4—24no—
44C2×C4×D4C2×C411yescharacteristic
45C2×C4×D4D412yes—
46C2×C4×D4D411yescharacteristic
47C2×C4×D4C2312yes—
48C2×#117—22no—
49C2×#117—21no—
50C2×#117—22no—
51C2×#117D411yescharacteristic
52C2×#117D412yes—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—21no—
55C22×C42C2×C412yes—
56C22×C42C2311yescharacteristic
57#431—22no—
58#431—22no—
59#431—22no—
60#431—22no—
61C2×C4⋊C8—22no—
62#410—44no—
63C2×#99—24no—
64C2×C22⋊C8C2314yes—
65C2×C4×C8—22no—
66#389—22no—
67#389—24no—
68#389—22no—
69#389—24no—
70#385—24no—
71#385—24no—
72#385—24no—
73#384—22no—
74#384—21no—
75#384—22no—
76#382—22no—
77#382—22no—
78#382—21no—
79#382—21no—
80#382—22no—
81#382C2×C412yes—
82#382C2×C414yes—
83#382C2×C411yescharacteristic
84#382C2×C412yes—
85#382C2×C411yescharacteristic
86#380—22no—
87#380—21no—
88#380—22no—
89#380C2×C414yes—
90#380C2314yes—
91#380C2312yes—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4C2×C412yes—
97C4×C4⋊C4C2×C414yes—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95C2×C414yes—
101C2×#95C2312yes—
102#342—24no—
103#324—24no—
104#323—24no—
105C2×#41—22no—
106C2×#41—22no—
107C2×#41—24no—
108C2×#41SD1614yes—
109C2×#41Q1614yes—
110C22×Q8C22⋊C411yescharacteristic
111C22×D4C22⋊C411yescharacteristic
112C22×D4C2×D412yes—
113C23×C4—24no—
114C23×C4—22no—
115C23×C4C22⋊C412yes—
116C23×C4C22⋊C412yes—
117C23×C4C22×C412yes—
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2411yescharacteristic
121C22×C8—24no—
122C4×Q8—22no—
123C4×Q8—22no—
124C4×Q8—22no—
125C4×Q8—22no—
126C4×Q8—48no—
127C4×D4—24no—
128C4×D4—24no—
129C4×D4—24no—
130C4×D4—24no—
131C4×D4—22no—
132C4×D4—22no—
133C4×D4—22no—
134C4×D4—22no—
135C4×D4—416no—
136C4×D4—48no—
137#117—22no—
138#117—22no—
139#117—24no—
140#117—24no—
141#117—48no—
142#117—44no—
143#117—44no—
144#117—44no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—48no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4C22⋊C411yescharacteristic
160C2×C4⋊C4C22⋊C412yes—
161C2×C4⋊C4C22⋊C411yescharacteristic
162C2×C4⋊C4C2×D412yes—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4C22⋊C412yes—
167C2×C22⋊C4C2×D412yes—
168C2×C22⋊C4C2×D412yes—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—24no—
174C2×C42—22no—
175C2×C42—42no—
176C2×C42C22⋊C412yes—
177C2×C42C22⋊C412yes—
178C2×C42C22×C412yes—
179C2×C42C22×C411yescharacteristic
180C2×C42C22×C411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42#4112yes—
184C2×C42#4112yes—
185C2×C42#4112yes—
186C2×C42#4112yes—
187C4⋊C8—42no—
188C4⋊C8—42no—
189#99—48no—
190C22⋊C8—24no—
191C22⋊C8—24no—
192C4×C8—42no—
193C4×C8—42no—
194#95—21no—
195#95—22no—
196#95—24no—
197#95—22no—
198#95—24no—
199#95—21no—
200#95—22no—
201#95—24no—
202#95—42no—
203#95C22×C412yes—
204#95C22×C414yes—
205#95C22×C412yes—
206C24C2×C22⋊C412yes—
207C24C22×D411yescharacteristic
208#41—28no—
209#41—28no—
210#41—28no—
211#41—28no—
212C2×Q8—22no—
213C2×Q8—22no—
214C2×Q8#10314yes—
215C2×D4—24no—
216C2×D4—24no—
217C2×D4—22no—
218C2×D4—22no—
219C2×D4#10314yes—
220C2×D4C2×SD1614yes—
221C2×D4C2×Q1614yes—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—24no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—48no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4—81no—
236C22×C4—81no—
237C22×C4—82no—
238C22×C4—82no—
239C22×C4—82no—
240C22×C4—82no—
241C22×C4—84no—
242C22×C4—84no—
243C22×C4#10318yes—
244C22×C4C2×C22⋊C412yes—
245C22×C4C2×C22⋊C412yes—
246C22×C4C2×C22⋊C412yes—
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4#12312yes—
252C22×C4#12312yes—
253C22×C4#12312yes—
254C22×C4#12312yes—
255C22×C4C2×SD1612yes—
256C22×C4C2×Q1612yes—
257C22×C4AGL(1,ℤ/8)14yes—
258C22×C4C23×C411yescharacteristic
259C22×C4C22×D411yescharacteristic
260C22×C4C2×#4112yes—
261C22×C4C2×#4112yes—
262C2×C8—44no—
263C2×C8—84no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—48no—
269C4⋊C4—42no—
270C4⋊C4—44no—
271C4⋊C4—44no—
272C4⋊C4—42no—
273C4⋊C4—82no—
274C4⋊C4—82no—
275C4⋊C4—84no—
276C4⋊C4—84no—
277C4⋊C4—84no—
278C4⋊C4—84no—
279C4⋊C4—84no—
280C22⋊C4—24no—
281C22⋊C4—24no—
282C22⋊C4—24no—
283C22⋊C4—48no—
284C22⋊C4—44no—
285C22⋊C4—44no—
286C22⋊C4—44no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—22no—
293C42—48no—
294C42—44no—
295C42—84no—
296C42#11711yescharacteristic
297C42#11711yescharacteristic
298C42#11711yescharacteristic
299C42#11711yescharacteristic
300C42#11712yes—
301C42#11712yes—
302C42#12312yes—
303C42#12312yes—
304C42#12312yes—
305C42#12312yes—
306C23—22no—
307C23—22no—
308C23—41no—
309C23—41no—
310C23—42no—
311C23—42no—
312C23C2×#10314yes—
313C23C22×C22⋊C411yescharacteristic
314C23C2×#12312yes—
315C23C22×SD1611yescharacteristic
316C23C22×Q1611yescharacteristic
317C23C2×AGL(1,ℤ/8)12yes—
318Q8—28no—
319D4—28no—
320D4—28no—
321D4—28no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—28no—
325C2×C4—28no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—84no—
352C2×C4—84no—
353C2×C4—84no—
354C2×C4—164no—
355C2×C4—168no—
356C2×C4#38111yescharacteristic
357C2×C4#38111yescharacteristic
358C2×C4#38112yes—
359C2×C4#38111yescharacteristic
360C2×C4#38111yescharacteristic
361C2×C4#38112yes—
362C2×C4C2×#10314yes—
363C2×C4C2×#10312yes—
364C2×C4C2×#10312yes—
365C2×C4#41812yes—
366C2×C4#41812yes—
367C2×C4#48114yes—
368C2×C4#48114yes—
369C2×C4#48414yes—
370C2×C4#48414yes—
371C2×C4C2×#11711yescharacteristic
372C2×C4C2×#11711yescharacteristic
373C2×C4C2×#12312yes—
374C8—84no—
375C22—21no—
376C22—22no—
377C22—21no—
378C22—22no—
379C22—24no—
380C22—42no—
381C22—41no—
382C22—41no—
383C22—44no—
384C22—44no—
385C22—42no—
386C22—42no—
387C22C2×#38111yescharacteristic
388C22C22×#10311yescharacteristic
389C22C2×#41811yescharacteristic
390C22C2×#48112yes—
391C22C2×#48412yes—
392C4—24no—
393C4—21no—
394C4—21no—
395C4—22no—
396C4—22no—
397C4—21no—
398C4—21no—
399C4—42no—
400C4—42no—
401C4—82no—
402C4—84no—
403C4—82no—
404C4—164no—
405C4#182412yes—
406C4#182412yes—
407C4#182414yes—
408C2—21no—
409C2—22no—
410C2—21no—
411C2—22no—
412C2—44no—
413C2C2×#182411yescharacteristic
414C2#2070811yescharacteristic
415C2#2071011yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #8761C2semidirect
5. #8026C2semidirect
6. #7981C2semidirect
12. #2351C4semidirect
15. #2349C4semidirect
19. #2285C4semidirect
22. C2×#382C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#341842
#206322
#173492
#87612
#80262
#79812